Logistic function from Lakshay Garg

Percentage Accurate: 54.5% → 99.9%

Time: 20.1s

Precision: binary64

Cost: 39240

Math

\[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]

↓

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -2:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} + -1\right)}^{3}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)}\right)\\ \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))

↓

(FPCore (x y)
 :precision binary64
 (if (<= (* -2.0 x) -2.0)
   (cbrt (pow (+ (/ 2.0 (+ (pow (exp x) -2.0) 1.0)) -1.0) 3.0))
   (if (<= (* -2.0 x) 0.002)
     (+
      (* -0.3333333333333333 (pow x 3.0))
      (+ x (* 0.13333333333333333 (pow x 5.0))))
     (log (exp (expm1 (- (log 2.0) (log1p (exp (* -2.0 x))))))))))

C

double code(double x, double y) {
	return (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
}

↓

double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -2.0) {
		tmp = cbrt(pow(((2.0 / (pow(exp(x), -2.0) + 1.0)) + -1.0), 3.0));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = (-0.3333333333333333 * pow(x, 3.0)) + (x + (0.13333333333333333 * pow(x, 5.0)));
	} else {
		tmp = log(exp(expm1((log(2.0) - log1p(exp((-2.0 * x)))))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	return (2.0 / (1.0 + Math.exp((-2.0 * x)))) - 1.0;
}

↓

public static double code(double x, double y) {
	double tmp;
	if ((-2.0 * x) <= -2.0) {
		tmp = Math.cbrt(Math.pow(((2.0 / (Math.pow(Math.exp(x), -2.0) + 1.0)) + -1.0), 3.0));
	} else if ((-2.0 * x) <= 0.002) {
		tmp = (-0.3333333333333333 * Math.pow(x, 3.0)) + (x + (0.13333333333333333 * Math.pow(x, 5.0)));
	} else {
		tmp = Math.log(Math.exp(Math.expm1((Math.log(2.0) - Math.log1p(Math.exp((-2.0 * x)))))));
	}
	return tmp;
}

Julia

function code(x, y)
	return Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0)
end

↓

function code(x, y)
	tmp = 0.0
	if (Float64(-2.0 * x) <= -2.0)
		tmp = cbrt((Float64(Float64(2.0 / Float64((exp(x) ^ -2.0) + 1.0)) + -1.0) ^ 3.0));
	elseif (Float64(-2.0 * x) <= 0.002)
		tmp = Float64(Float64(-0.3333333333333333 * (x ^ 3.0)) + Float64(x + Float64(0.13333333333333333 * (x ^ 5.0))));
	else
		tmp = log(exp(expm1(Float64(log(2.0) - log1p(exp(Float64(-2.0 * x)))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

↓

code[x_, y_] := If[LessEqual[N[(-2.0 * x), $MachinePrecision], -2.0], N[Power[N[Power[N[(N[(2.0 / N[(N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision], If[LessEqual[N[(-2.0 * x), $MachinePrecision], 0.002], N[(N[(-0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(0.13333333333333333 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[Exp[N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 -2 x) < -2

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]

   2. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}\right)}^{2}} \]
      Step-by-step derivation

      [Start] 100.0%	\[ \frac{2}{1 + e^{-2 \cdot x}} - 1 \]
      add-sqr-sqrt [=>] 100.0%	\[ \color{blue}{\sqrt{\frac{2}{1 + e^{-2 \cdot x}} - 1} \cdot \sqrt{\frac{2}{1 + e^{-2 \cdot x}} - 1}} \]
      pow2 [=>] 100.0%	\[ \color{blue}{{\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)}^{2}} \]
      add-exp-log [=>] 100.0%	\[ {\left(\sqrt{\color{blue}{e^{\log \left(\frac{2}{1 + e^{-2 \cdot x}}\right)}} - 1}\right)}^{2} \]
      expm1-def [=>] 100.0%	\[ {\left(\sqrt{\color{blue}{\mathsf{expm1}\left(\log \left(\frac{2}{1 + e^{-2 \cdot x}}\right)\right)}}\right)}^{2} \]
      log-div [=>] 100.0%	\[ {\left(\sqrt{\mathsf{expm1}\left(\color{blue}{\log 2 - \log \left(1 + e^{-2 \cdot x}\right)}\right)}\right)}^{2} \]
      log1p-udef [<=] 100.0%	\[ {\left(\sqrt{\mathsf{expm1}\left(\log 2 - \color{blue}{\mathsf{log1p}\left(e^{-2 \cdot x}\right)}\right)}\right)}^{2} \]
      exp-prod [=>] 100.0%	\[ {\left(\sqrt{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(\color{blue}{{\left(e^{-2}\right)}^{x}}\right)\right)}\right)}^{2} \]

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto {\left(\sqrt{\mathsf{expm1}\left(\color{blue}{\log 2 - \log \left(1 + e^{-2 \cdot x}\right)}\right)}\right)}^{2} \]

   4. Simplified 100.0%

      \[\leadsto {\left(\sqrt{\mathsf{expm1}\left(\color{blue}{\log 2 - \mathsf{log1p}\left(e^{x \cdot -2}\right)}\right)}\right)}^{2} \]
      Step-by-step derivation

      [Start] 100.0%	\[ {\left(\sqrt{\mathsf{expm1}\left(\log 2 - \log \left(1 + e^{-2 \cdot x}\right)\right)}\right)}^{2} \]
      log1p-def [=>] 100.0%	\[ {\left(\sqrt{\mathsf{expm1}\left(\log 2 - \color{blue}{\mathsf{log1p}\left(e^{-2 \cdot x}\right)}\right)}\right)}^{2} \]
      *-commutative [=>] 100.0%	\[ {\left(\sqrt{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{\color{blue}{x \cdot -2}}\right)\right)}\right)}^{2} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} + -1\right)}^{3}}} \]
      Step-by-step derivation

      [Start] 100.0%	\[ {\left(\sqrt{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{x \cdot -2}\right)\right)}\right)}^{2} \]
      unpow2 [=>] 100.0%	\[ \color{blue}{\sqrt{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{x \cdot -2}\right)\right)} \cdot \sqrt{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{x \cdot -2}\right)\right)}} \]
      add-sqr-sqrt [<=] 100.0%	\[ \color{blue}{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{x \cdot -2}\right)\right)} \]
      add-cbrt-cube [=>] 100.0%	\[ \color{blue}{\sqrt[3]{\left(\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{x \cdot -2}\right)\right) \cdot \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{x \cdot -2}\right)\right)\right) \cdot \mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{x \cdot -2}\right)\right)}} \]
      pow3 [=>] 100.0%	\[ \sqrt[3]{\color{blue}{{\left(\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{x \cdot -2}\right)\right)\right)}^{3}}} \]
      expm1-udef [=>] 100.0%	\[ \sqrt[3]{{\color{blue}{\left(e^{\log 2 - \mathsf{log1p}\left(e^{x \cdot -2}\right)} - 1\right)}}^{3}} \]
      sub-neg [=>] 100.0%	\[ \sqrt[3]{{\color{blue}{\left(e^{\log 2 - \mathsf{log1p}\left(e^{x \cdot -2}\right)} + \left(-1\right)\right)}}^{3}} \]
      exp-diff [=>] 100.0%	\[ \sqrt[3]{{\left(\color{blue}{\frac{e^{\log 2}}{e^{\mathsf{log1p}\left(e^{x \cdot -2}\right)}}} + \left(-1\right)\right)}^{3}} \]
      add-exp-log [<=] 100.0%	\[ \sqrt[3]{{\left(\frac{\color{blue}{2}}{e^{\mathsf{log1p}\left(e^{x \cdot -2}\right)}} + \left(-1\right)\right)}^{3}} \]
      log1p-udef [=>] 100.0%	\[ \sqrt[3]{{\left(\frac{2}{e^{\color{blue}{\log \left(1 + e^{x \cdot -2}\right)}}} + \left(-1\right)\right)}^{3}} \]
      add-exp-log [<=] 100.0%	\[ \sqrt[3]{{\left(\frac{2}{\color{blue}{1 + e^{x \cdot -2}}} + \left(-1\right)\right)}^{3}} \]
      +-commutative [=>] 100.0%	\[ \sqrt[3]{{\left(\frac{2}{\color{blue}{e^{x \cdot -2} + 1}} + \left(-1\right)\right)}^{3}} \]
      exp-prod [=>] 100.0%	\[ \sqrt[3]{{\left(\frac{2}{\color{blue}{{\left(e^{x}\right)}^{-2}} + 1} + \left(-1\right)\right)}^{3}} \]
      metadata-eval [=>] 100.0%	\[ \sqrt[3]{{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} + \color{blue}{-1}\right)}^{3}} \]

   if -2 < (*.f64 -2 x) < 2e-3

   1. Initial program 9.2%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot {x}^{3} + \left(0.13333333333333333 \cdot {x}^{5} + x\right)} \]

   if 2e-3 < (*.f64 -2 x)

   1. Initial program 99.9%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]

   2. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\log \left(e^{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{-2}\right)}^{x}\right)\right)}\right)} \]
      Step-by-step derivation

      [Start] 99.9%	\[ \frac{2}{1 + e^{-2 \cdot x}} - 1 \]
      add-log-exp [=>] 99.9%	\[ \color{blue}{\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)} \]
      add-exp-log [=>] 99.9%	\[ \log \left(e^{\color{blue}{e^{\log \left(\frac{2}{1 + e^{-2 \cdot x}}\right)}} - 1}\right) \]
      expm1-def [=>] 99.9%	\[ \log \left(e^{\color{blue}{\mathsf{expm1}\left(\log \left(\frac{2}{1 + e^{-2 \cdot x}}\right)\right)}}\right) \]
      log-div [=>] 99.9%	\[ \log \left(e^{\mathsf{expm1}\left(\color{blue}{\log 2 - \log \left(1 + e^{-2 \cdot x}\right)}\right)}\right) \]
      log1p-udef [<=] 100.0%	\[ \log \left(e^{\mathsf{expm1}\left(\log 2 - \color{blue}{\mathsf{log1p}\left(e^{-2 \cdot x}\right)}\right)}\right) \]
      exp-prod [=>] 100.0%	\[ \log \left(e^{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(\color{blue}{{\left(e^{-2}\right)}^{x}}\right)\right)}\right) \]

   3. Taylor expanded in x around inf 99.9%

      \[\leadsto \log \left(e^{\mathsf{expm1}\left(\color{blue}{\log 2 - \log \left(1 + e^{-2 \cdot x}\right)}\right)}\right) \]

   4. Simplified 100.0%

      \[\leadsto \log \left(e^{\mathsf{expm1}\left(\color{blue}{\log 2 - \mathsf{log1p}\left(e^{x \cdot -2}\right)}\right)}\right) \]
      Step-by-step derivation

      [Start] 0.0%	\[ {\left(\sqrt{\mathsf{expm1}\left(\log 2 - \log \left(1 + e^{-2 \cdot x}\right)\right)}\right)}^{2} \]
      log1p-def [=>] 0.0%	\[ {\left(\sqrt{\mathsf{expm1}\left(\log 2 - \color{blue}{\mathsf{log1p}\left(e^{-2 \cdot x}\right)}\right)}\right)}^{2} \]
      *-commutative [=>] 0.0%	\[ {\left(\sqrt{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{\color{blue}{x \cdot -2}}\right)\right)}\right)}^{2} \]

3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -2:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} + -1\right)}^{3}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	39240

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -2:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} + -1\right)}^{3}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(e^{-2 \cdot x}\right)\right)}\right)\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	26436

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -2:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} + -1\right)}^{3}}\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \log \left(e^{\frac{2}{1 + e^{-2 \cdot x}}}\right)\\ \end{array} \]

Alternative 3
Accuracy	99.9%
Cost	20296

\[\begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{if}\;-2 \cdot x \leq -2:\\ \;\;\;\;-1 + t_0\\ \mathbf{elif}\;-2 \cdot x \leq 0.002:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \log \left(e^{t_0}\right)\\ \end{array} \]

Alternative 4
Accuracy	99.9%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -2 \lor \neg \left(-2 \cdot x \leq 0.002\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot {x}^{3} + \left(x + 0.13333333333333333 \cdot {x}^{5}\right)\\ \end{array} \]

Alternative 5
Accuracy	99.9%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \leq -0.005 \lor \neg \left(-2 \cdot x \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;-1 + \frac{2}{1 + e^{-2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \end{array} \]

Alternative 6
Accuracy	99.7%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.16:\\ \;\;\;\;x + -0.3333333333333333 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Accuracy	99.4%
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Accuracy	75.8%
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Accuracy	27.6%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023263 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))